The dephasing rate SP formula
Encyclopedia
The SP formula for the dephasing
rate
of a particle that moves in a fluctuating environment, unifies various results that have been obtained, notably in condensed matter physics
with regard to the motion of electrons in a metal
.
The general case requires to take into account not only the temporal correlations but also the spatial correlations of the environmental fluctuations
.
These can be characterized by the spectral form factor
, while the motion of the particle is characterized by its power spectrum 
. Consequently at finite temperature the expression for the dephasing rate takes the following form that involves "S" and "P" functions:
Due to inherent limitations of the semiclassical (stationary phase) approximation the physically correct procedure is to use the non-symmetrized quantum versions of
and 
. The argument is based on the analogy of the above expression with Fermi-golden-rule
calculation of the transitions which are induced by the system-environment interaction.
, which describes motion in dynamical disorder. In order to derive the dephasing rate formula from first principles, a purity-based definition of the dephasing factor can be adopted.
The purity
describes how a quantum state becomes mixed due to the entanglement of the system with the environment. Using perturbation theory, one recovers at finite temperatures at the long time limit 
where the decay constant is given by the dephasing rate formula with non symmetrized spectral functions as expected. There is a somewhat controversial possibility to get power law decay of 
at the limit of zero temperature.
The proper way to incorporate Pauli blocking in the many-body dephasing calculation,
within the framework of the SP formula approach, has been clarified as well.
with temperature
and friction 
,
the spectral form factor is
This expression reflects that in the classical limit
the electron experiences "white temporal noise",
which means force that is not correlated in time,
but uniform is space (high
components are absent).
In contrast to that, for diffusive motion of an electron
in a 3D metallic environment, which is created by the rest of the electrons,
the spectral form factor is
This expression reflects that in the classical limit
the electron experiences "white spatio-temporal noise",
which means force that is neither correlated in time nor in space.
The power spectrum of a single diffusive electron is
But in the many body context this expression
acquires a "Fermi blocking factor":
Calculating the SP integral we get the well known result
.
Dephasing
Dephasing is a name for the mechanism that recovers classical behavior from a quantum system. It is an important effect in condensed matter physics, particularly in the study of mesoscopic devices...
rate
of a particle that moves in a fluctuating environment, unifies various results that have been obtained, notably in condensed matter physicsCondensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
with regard to the motion of electrons in a metal
.
The general case requires to take into account not only the temporal correlations but also the spatial correlations of the environmental fluctuations
.
These can be characterized by the spectral form factor
, while the motion of the particle is characterized by its power spectrum . Consequently at finite temperature the expression for the dephasing rate takes the following form that involves "S" and "P" functions:Due to inherent limitations of the semiclassical (stationary phase) approximation the physically correct procedure is to use the non-symmetrized quantum versions of
and . The argument is based on the analogy of the above expression with Fermi-golden-ruleFermi's golden rule
In quantum physics, Fermi's golden rule is a way to calculate the transition rate from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to a perturbation....
calculation of the transitions which are induced by the system-environment interaction.
Derivation
It is most illuminating to understand the SP formula in the context of the DLD modelDissipation model for extended environment
A unified model for Diffusion Localization and Dissipation , optionally termed Diffusion with Local Dissipation, has been introduced for the study of Quantal Brownian Motion in dynamical disorder....
, which describes motion in dynamical disorder. In order to derive the dephasing rate formula from first principles, a purity-based definition of the dephasing factor can be adopted.
The purity
describes how a quantum state becomes mixed due to the entanglement of the system with the environment. Using perturbation theory, one recovers at finite temperatures at the long time limit where the decay constant is given by the dephasing rate formula with non symmetrized spectral functions as expected. There is a somewhat controversial possibility to get power law decay of at the limit of zero temperature.The proper way to incorporate Pauli blocking in the many-body dephasing calculation,
within the framework of the SP formula approach, has been clarified as well.
Example
For the standard 1D Caldeira-Leggett Ohmic environment,with temperature
and friction ,the spectral form factor is
This expression reflects that in the classical limit
the electron experiences "white temporal noise",
which means force that is not correlated in time,
but uniform is space (high
components are absent).In contrast to that, for diffusive motion of an electron
in a 3D metallic environment, which is created by the rest of the electrons,
the spectral form factor is
This expression reflects that in the classical limit
the electron experiences "white spatio-temporal noise",
which means force that is neither correlated in time nor in space.
The power spectrum of a single diffusive electron is
But in the many body context this expression
acquires a "Fermi blocking factor":
Calculating the SP integral we get the well known result
.